Finite Group Databasebeta

← #26125552

Subgroups — #26125552

Order 12032.

460 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20241712C212yesmaximal, complemented
2#20241332C212yesmaximal, complemented
3#20241320C212yesmaximal
4C4×#124×C47C211yesmaximal, characteristic
5#15689176—22no—
6#15688849—22no—
7#15688849C2214yes—
8C2×#124×C47C411yescharacteristic
9C2×C4×Q8×C47C2211yescharacteristic
10C2×C4×D4×C47C411yescharacteristic
11C47×#391C411yescharacteristic
12C47×#390C2211yescharacteristic
13C47×#388—22no—
14C47×#382C2211yescharacteristic
15C4×C22⋊C4×C47—22no—
16C43×C47C411yescharacteristic
17C4⋊(C47⋊C8)—42no—
18C22⋊(C47⋊C8)—24no—
19C4×C47⋊C8—42no—
20C22×Q8×C47C2×C411yescharacteristic
21C22×D4×C47C2×C411yescharacteristic
22C23×C4×C47—21no—
23#124×C47—22no—
24#124×C47C814yescomplemented
25C4×Q8×C47—41no—
26C4×Q8×C47—41no—
27C4×D4×C47—41no—
28C4×D4×C47—41no—
29C2×C4⋊C4×C47C2×C411yescharacteristic
30C2×C4⋊C4×C47D412yes—
31C2×C4⋊C4×C47C2311yescharacteristic
32C2×C22⋊C4×C47—22no—
33C2×C22⋊C4×C47—21no—
34C2×C22⋊C4×C47—22no—
35C2×C42×C47—21no—
36C2×C42×C47—21no—
37C2×C42×C47C2×C411yescharacteristic
38C2×C42×C47C2×C411yescharacteristic
39C2×C42×C47C2×C411yescharacteristic
40C2×C42×C47D412yes—
41#95×C47—22no—
42#95×C47—22no—
43C24×C47—21no—
44C2×Q8×C47—41no—
45C2×Q8×C47—41no—
46C2×Q8×C47C2×C812yes—
47C2×Q8×C47M1612yes—
48C2×D4×C47—21no—
49C2×D4×C47—41no—
50C2×D4×C47—41no—
51C2×D4×C47C2×C812yes—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—44no—
56C22×C4×C47—41no—
57C22×C4×C47—41no—
58C22×C4×C47C22⋊C412yes—
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D412yes—
63C4⋊C4×C47—42no—
64C4⋊C4×C47—41no—
65C4⋊C4×C47—41no—
66C22⋊C4×C47—28no—
67C22⋊C4×C47—44no—
68C22⋊C4×C47—42no—
69C42×C47—21no—
70C42×C47—21no—
71C42×C47—22no—
72C42×C47—22no—
73C42×C47—21no—
74C42×C47—42no—
75C42×C47—42no—
76C42×C47—41no—
77C42×C47C2×C812yes—
78C2×C47⋊C8—44no—
79C23×C47—22no—
80C23×C47—42no—
81C23×C47—42no—
82C23×C47—41no—
83C23×C47—41no—
84C23×C47C2×C22⋊C411yescharacteristic
85Q8×C47—42no—
86Q8×C47—42no—
87D4×C47—42no—
88D4×C47—42no—
89C2×C4×C47—21no—
90C2×C4×C47—22no—
91C2×C4×C47—21no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—21no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—41no—
106C2×C4×C47—82no—
107C2×C4×C47—82no—
108C2×C4×C47C22⋊C814yes—
109C2×C4×C47#9911yescharacteristic
110C2×C4×C47#9911yescharacteristic
111C2×C4×C47C22×C811yescharacteristic
112C2×C4×C47C2×M1611yescharacteristic
113C47⋊C8—84no—
114#11476—471nomaximal, Sylow
115C22×C47—21no—
116C22×C47—21no—
117C22×C47—44no—
118C22×C47—42no—
119C22×C47—82no—
120C22×C47—81no—
121C22×C47—81no—
122C22×C47C2×C22⋊C811yescharacteristic
123C22×C47C2×#9911yescharacteristic
124C22×C47#41311yescharacteristic
125C4×C47—21no—
126C4×C47—24no—
127C4×C47—21no—
128C4×C47—21no—
129C4×C47—42no—
130C4×C47—42no—
131C4×C47—42no—
132C4×C47—41no—
133C4×C47—81no—
134C4×C47#35512yes—
135C4×#124D4711yescharacteristic
136#1461—472no—
137#1324—472no—
138#1318—472no—
139C2×C47—21no—
140C2×C47—21no—
141C2×C47—82no—
142C2×C47#145911yescharacteristic
143C2×C47#212611yescharacteristic
144C2×C47C2×#35511yescharacteristic
145C2×#124Dic4711yescharacteristic
146C2×C4×Q8C2×D4711yescharacteristic
147C2×C4×D4Dic4711yescharacteristic
148#431—942no—
149#391Dic4711yescharacteristic
150#390C2×D4711yescharacteristic
151#388—22no—
152#382C2×D4711yescharacteristic
153C4×C22⋊C4—22no—
154C43Dic4711yescharacteristic
155#324—474no—
156#324—942no—
157C47#1147611yescharacteristic, complemented, Sylow
158C22×Q8C2×Dic4711yescharacteristic
159C22×D4C2×Dic4711yescharacteristic
160C23×C4—21no—
161#124—22no—
162#124C47⋊C814yescomplemented
163C4×Q8—41no—
164C4×Q8—41no—
165C4×D4—41no—
166C4×D4—41no—
167C2×C4⋊C4C2×Dic4711yescharacteristic
168C2×C4⋊C4C22⋊D4712yes—
169C2×C4⋊C4C22×D4711yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42C2×Dic4711yescharacteristic
176C2×C42C2×Dic4711yescharacteristic
177C2×C42C2×Dic4711yescharacteristic
178C2×C42C22⋊D4712yes—
179C4⋊C8—1882no—
180C22⋊C8—944no—
181C4×C8—1882no—
182#95—22no—
183#95—22no—
184C24—21no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8C2×C47⋊C812yes—
188C2×Q8#1062532912yes—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4C2×C47⋊C812yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C22⋊Dic4712yes—
200C22×C4C22⋊Dic4711yescharacteristic
201C22×C4C22⋊Dic4711yescharacteristic
202C22×C4C22×Dic4711yescharacteristic
203C22×C4C2×C22⋊D4712yes—
204C2×C8—1884no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C22⋊C4—28no—
209C22⋊C4—44no—
210C22⋊C4—42no—
211C42—22no—
212C42—21no—
213C42—22no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—42no—
218C42—42no—
219C42C2×C47⋊C812yes—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23C2×C22⋊Dic4711yescharacteristic
226Q8—42no—
227Q8—42no—
228D4—42no—
229D4—42no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4C22⋊(C47⋊C8)14yes—
250C2×C4#1309393911yescharacteristic
251C2×C4#1309393911yescharacteristic
252C2×C4C22×C47⋊C811yescharacteristic
253C2×C4C2×#1062532911yescharacteristic
254C8—3764no—
255C22—21no—
256C22—21no—
257C22—44no—
258C22—42no—
259C22—82no—
260C22—81no—
261C22—81no—
262C22C2×C22⋊(C47⋊C8)11yescharacteristic
263C22C2×#1309393911yescharacteristic
264C22#1568910711yescharacteristic
265C4—24no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—42no—
272C4—41no—
273C4—81no—
274C4#1568892012yes—
275C2—21no—
276C2—21no—
277C2—82no—
278C2#2024170611yescharacteristic
279C2#2024505311yescharacteristic
280C2C2×#1568892011yescharacteristic
subgroupG/H
centerC2×C4#13093939
commutatorC2×C4×C47C22×C8
FrattiniC2×C4⋊C4C22×D47
FittingC4×#124×C47C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC2×C4⋊C4×C47C23
hypercenterC4×#124D47
hypocenterC47#11476
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#124

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20241712C2semidirect
2. #20241332C2semidirect
24. #124×C47C8semidirect
157. C47#11476semidirect
162. #124C47⋊C8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202417122
#202413322
#202413202
C4×#124×C472
#1147647